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 Incoming links: Second-order fluid

The local motion is simple shear with flow direction

z

, gradient direction

r

, and vorticity direction

θ

q = d u / d r

N_{1} = τ_{zz} - τ_{rr} = Ψ_{1} q^{2}, N_{2} = τ_{rr} - τ_{θθ} = Ψ_{2} q^{2} .

Up to an isotropic contribution absorbed into pressure, a convenient representation is

τ_{zz} = (Ψ_{1} + Ψ_{2}) q^{2}, τ_{rr} = Ψ_{2} q^{2}, τ_{θθ} = 0.

Thus both the flow-direction and gradient-direction normal stresses can be nonzero, while the radial momentum equation becomes

\frac{d p}{d r} = \frac{1}{r} \frac{d ( r τ _{rr} )}{d r} - \frac{τ _{θθ}}{r} = \frac{Ψ _{2}}{r} \frac{d}{d r} (r q^{2}) .

τ_{zz}

but cancels from radial balance.

In the yielded layer,

q = \frac{1}{η} [\frac{G}{2} (\frac{b ^{2}}{r} - r) - σ_{y}] > 0.

Moreover,

\frac{d}{d r} (r q^{2}) = \frac{q}{η} [- \frac{G}{2} (\frac{b ^{2}}{r} + 3 r) - σ_{y}] < 0.

Consequently

d p / d r

has the sign opposite to

Ψ_{2}

and is independent of

Ψ_{1}

. For the common case

Ψ_{2} < 0

, pressure increases radially outward toward the free surface.